reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th19:
  for n being non zero Nat holds
  Euler_factorization (p|^n) = p .--> (p|^n-p|^(n-1))
  proof
    let n be non zero Nat;
    set x = p|^n-p|^(n-1);
    set f = Euler_factorization (p|^n);
    set g = p .--> x;
A1: dom f = support ppf (p|^n) by Def1
    .= support pfexp (p|^n) by NAT_3:def 9
    .= {p} by NAT_3:42;
A2: dom g = {p};
A3: p |-count (p|^n) = n by NAT_3:25,INT_2:def 4;
    p in {p} by TARSKI:def 1;
    then
    consider c being non zero Nat such that
A4: c = p |-count (p|^n) & f.p = p|^c - p|^(c-1) by A1,Def1;
    now
      let a be object;
      assume a in dom f;
      then a = p by A1,TARSKI:def 1;
      hence f.a = g.a by A3,A4,FUNCOP_1:72;
    end;
    hence thesis by A1,A2,FUNCT_1:def 11;
  end;
